When relaxing the constraint in (REF ), the problem still requires to be solved on \(\mathcal {S}^{++}_p\) .
Since \(\mathcal {S}^{++}_p\)  is open in \(\mathcal {S}_p\)  the tangent space \(T_{{\Sigma }}\mathcal {S}^{++}_p\)  can simply be identified as \(\mathcal {S}_p\) , for any \({\Sigma }\in \mathcal {S}^{++}_p\) .
To be able to perform optimization, the first step is to define a Riemannian metric, i.e., an inner product on this tangent space.
In the case of \(\mathcal {S}^{++}_p\) , the most natural choice is the affine-invariant metric [1]}, [2]}, that corresponds to the Fisher information metric of the Gaussian distribution and features very interesting properties from a geometrical point of view.
It is defined for all \({\Sigma }\in \mathcal {S}^{++}_p\) , \({\xi }, \, {\eta }\in \mathcal {S}_p\)  as
\(\langle {\xi },{\eta }\rangle ^{\mathcal {S}^{++}_p}_{{\Sigma }} = \operatorname{tr}({\Sigma }^{-1}{\xi }{\Sigma }^{-1}{\eta }).\) 
